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jt/ ) ELECTROMECHA~CAL

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AniSlant Profeuor of Electrical Engineering

Profeuor of Electrical Engineering V"

The Monachuse"s Institute of Technology

Deportment of Electricol Engineering

HERBERT H. WOODSON

/...

DAVID C. WHITE

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This book is one of several resulting from a recent reVISIon of the 1Electrical Engineering Course at The Massachusetts Institute of Tech::/l'"

nology. Thc books have the general format of texts and are being used as such. However, they might well be described as reports on a research program aimed at the evolution of an undergraduate core curriculum in Electrical Engineering that will form a basis for a continuing career

.. in a field that is ever-changing. The development of an educational program in Electrical Engineering ,

to keep pace with the changes in technology is not a new endeavor at ,

" The Massachusetts Institute of Technology. In the early 1930's, the Faculty of the Department undertook a major review and reassessment of its program. By 1940, a series of new courses had been evolved,

("

.... and resulted in the publication of four rebted hooks,

The new technology that appeared during World War II brought great change to the field of Electrical Engincering. In recognition of this ... "\ fact, the Faculty of the Department undertook another reassessment of its program. By about 1952, a pattern for a curriculum had been evolved and its implementation was initiated with a high degree ofTS; ", ;~! J"4 -".,'~

enthusiasm and vigor. ~'f)."- ~;;j t.t'

The new curriculum subordinates option structures built around areas of industrial practice in favor of a common core that provides a broad

t "('tiM!" base for the engineering applications of the sciences. This core structure

11, includes a newly developed laboratory program which stresses thl: role of experimentation and its relation to theoretical model-making in the

.))'

solution of engineering problems. Faced with the time limitation of a four-year program for the Bachelor's degree, the entire core curriculum gives priority to basic principles and methods of analysis rather than ~~'"

to the presentation of current technology. J. A. STRATTON

yii

i

j.

~"

" Ii P R E F A C E" ';;;

~

'Jill'

PREFACEx J' sound base on which to build an understanding of electromechanical interactions. Although such an approach is useful for solving some pre~cnt-day problems. it does not give adequate accent to the more general problem of electromechanical interactions.

This book has been strongly influenced by three sources. The initial impetus and major driving force was the formulation of a required, onesemester. senior course in electromechanical energy conversion that would treat in some depth fundamental concepts and at the same time treat specific transducers in sufficient generality that physical insight into transducer dynamics could be obtained. It was felt that with the everincreasing use of more accurate feedback control systems, the engineer must be aware of the dynamic properties of the transducers that he is substituting for human muscle. The second source of information was research sponsored by the U. S. Air Force in which the dynamic behavior of aircraft generating systems was studied. The third source was graduate teaching and thesis research.

Since electromechanical energy conversion can occur only by the interaction of electromagnetic fields and material bodies in motion, it is reasonable to begin with a treatment of macroscopic electrodynamics. We have experimented with this starting point in our senior course and find that a student who masters this topic has excellent preparation for proceeding to more advanced work. But we also have found that for the average student the time required for an adequate coverage of electrodynamics leaves little time for a treatment of lumped-parameter systems to the degree necessary if dynamic behavior is to be studied. Furthermore, it is in general impractical to describe the general dynamic behavior of energy conversion systems only in terms of fields. These.. considerations, plus the presence in our curriculum of a junior course in fields, energy, and forces which introduces electrodynamics, prompted us to start with a lumped-parameter approach with some references to field theory as the basis for determining when lumped parameters can be used and in defining the system parameters.

Having chosen the lumped-parameter approach, we chose to make the derivation of equations of motion in Chapter 1 quite general, the principal limitation being that the coupling system (Le., energy storage in fields) be conservative. Two approaches have been used: the method of arbitrary displacement and conservation of energy, and Hamilton's principle leading to Lagrange's equations. The first approach yields only electromechanical coupling terms; consequently, the equations of motion of a system must be written using force laws (Kirchhoff's laws and d'Alembert's principle), and much bookkeeping must be done in

PREFACE xi

complicated problems. The Lagrangian formulation, on the other hand, can be generalized to include the nonconservative parts or the system, so all bookkeeping is done automatically. Some educators may feel that the use of the Lagrangian formulation may tend to make crank-turners out of the students; however, we feel that the Lagrangian has further significance than just being a tool for solving problems. It introduces students to variational principles which are in one sense as fundamental as the conservation principles. The Lagrangian formulation also introduces generalized coordinates and lays a firm foundation for the meaning of independent variables. The Lagrangian state function and the Legendre transformation provide an entry into state functions in general and a tie with state functions commonly used in thermodynamics. We feel that these advantages ot" the Lagrangian formulation make its introduction well worthwhile. This is in line with the desire to broaden the scope of the present treatment and to lay a foundation which in the future can be expanded to include systems other than those with only electrical and mechanical in teractions.

Chapter 2 treats mathematical techniques for solving equations of motion, starting with the straightforward classical solution of linear differential equations with constant coefficients and ending with the use of analog computers to solve nonlinear equations. As is obvious from the content, the purpose of this chapter is not to teach mathematics but to illustrate the use of known techniques in analyzing typical transducers. No attempt has been made to include all the known methods for analyzing linear equations. Notable omissions are electrical analogs, flow graphs, and operational tcchniques other than Laplace transforms. We feel that the techniques used are representative. No examples of the use of digital computers are included. However, we recognize the increasing importance of digital computation in system analysis and the general equations derived for electromechanical energy convertors are directly applicable to digital computation.

Chapters 3 and 4 represent an errort to integrate the earlier work t,,: of Kron, Gibbs, and others into a unified treatment of the dynamics of

I' !;, ~ '~.' rotating machines. By defining a general physical model consisting

~ 7, 'l)' of concentric magnetic cylinders with current sheets on their surfaces, {,\ (. a field solution is obtained, parameters are defined, and equations of

motion are derived. Starting from a single model, the complete dynamic I~ . I '. equations of most conventional machines and many unconventional ones

;~i; are obtained by selected constraints to the general model. Transforma '... tions are introduced that mathematically describe the physical change of variables made by a commutator and put the equations of motion in

I~ ~ ; ~ ~ \,~, ; I",; ;, ., ..

_-----oLL

xiii xii PREFACE

PREFACE

IJ.'. ,ii,

'" ' ,,':!i.

'. ~

.. .~.;",I},~,;:,

'i

".

more recognizahle and solvable forms. Thus one physical model with one set of equations dcsnibes all of conventional steady-state machine theory, the dynamic behavior of conventional machines, and the steadystate and dynamic characteristics of many unconventional machines.

The usc uf the generalized approach with the aid uf a Iahoratory machine having the same generality reduces steady-state machine theory to the solution of steady-state a-c and d-c circuit prohlems with which our students arc fully familiar. Furthermore, the general field solution allows interpret:1tion of energy conversion properties in terms of fields. Since so little time is required for this aspect of machine theory, much time remains for study of the dynamic properties which are so important in the engineering of today and which will become more important in the future.

There may he some question about the choice of a two-phase machine as the deviL'e for teaching machine theory. It is shown in Chapter 10 that the energy conversion characteristics of any polyphase machine with symmetrical impedances can be obtained from an equivalent two[,hase machine for which the equations of motion arc ohtained by a str:llghtforward change of variables. We feel that the mathematical cUll1pkxily of L'ven a two phase systcm tends to ohscurc many of the concepts contained in the treatment. The use of a three-phase o;ystem as the analytical vehicle only introduces additional mathematical complexities and further detracts from the understanding of the nature of energy conversion in rotating machines.

Chapter 5 gives a simple introduction to feedback control system theory. We feel that such an inclusion is desirable because those students who do not take such a course as an elective should nevertheless be exposed to the essential ideas involved. In addition, this chapter illustrates through an example how the dynamic behavior of an interconnected system of machines is affected by the characteristics of the machines.

Chapters I through 5 contain the material prepared for a first-term senior course. More recently, it is being taught as a second-term junior,; course. The classroom work is augmented by a laboratory in which transducers and the generalized machine are used to get the fundamental concepts across. In addition, commercial machines and transducers are used in experiments that stress dynamics, interconnected systems, and feedback control theory.

Chapters 6 through 9 are detailed and specialized treatments of specific devices. Their purpose is to illustrate techniques and to provide

t

t

j I !

information on applications of the general approach of the earlier char: ters. This material is primarily intended for use in graduate courses and for the use of practicing engineers. It has also been used as part of a one-semester senior elective subject on electric machine systems.

Chapters 1() and 1 I are necessary for generality in the treatment. Chapter IO.provides the analytical methods for reducing any n-m phase machine to an equivalent two-phase machine for considerations of dynamic energy c(lnversion properties and thus justifies the lise of the two-phase model in the carlier treatment. Chapter 11 shows how the results of the analysis of Chapter 3 with sinusoidal currcn t shcL:ts Can be extended to analyze a machine with current sheets (windings) of any arbitrary distrihution. It also contains a rigorous justification for the concept of analyzing a commutator by a simple transformation of variables. This one concept is of course vital to the establishment of the unity among aIt machine types.

The topics in this book were first presented to a group of seniors at MIT in the fall of 1954, and the material prepared at that time was a group efTort. It h:1S sincL: gonc through several modifications, but lllany of the basic ideas around which this text is formed were estahlished by the initial group, which consisted of the authors plus Professors MahmOUd Riaz, Robert M. Saunders, I-Jerman Koenig, and Richard H. Fnlzier. In addition to the initial group effort, special recognition is due to Professor Mahmoud Riaz for his contribution to Chapters 4, 6, 8, and 10; to Professors Leonard A. Gould and Robert M. Saunders for their contribution to Chapter 5; and to David Bobroff for his contribution to Chapter 1. Others who have made many contributions to the effort have been Professors Richard B. Adler, Lan J. Chu, David J. Epstein, _ Robert M. Fano, Charles Kingsley, Jr., Alexander Kusko, Osman K. Mawardi, Norman H. Meyers, and Karl L. Wildes. Special mention should also be made of the very excellent works of Drs. Gabriel Kron and W. J. Gibbs, whose very fine publications in this field were an invaluable aid to the development of many of the concepts presented here.

The authors are also greatly indebted to Bernard Lovell, whose diligence in editing and checking derivations was invaluable. We wish to. thank also the Misses Lucia Hunt, Ruth Coughlan, Lydia Bonazzoli, and Evelyn Fraccastoro for their perseverance in typing the drafts of the manuscript.

Special mention is due our wives Glorianna G. White and Blanche S. Woodson for the invaluable assistance, both tangible and intangible, given during the writing of this text.

Above all, the authors wish to express their appreciation for the

.. , ifV"\1'

xiv PREFACE

dynamic leadership and stimulation provided hy Dr. Gordon S. Brown, c o N T E N T s

without whme foresight, perseverance, and courage this book would not

have been possible. DAVID C. WHITE HERBERT H. WOODSON

Cambridge, Mass. November, 1958

n;

l.;

11~

The Dynamic Equations of Motion of Electromechanical,riU Chapter 1 Systems

2 Analytical Techniques for Treating Electromechanical Equa

tions of Motion Including Typical Transducers as Examples 87

. 3 The Generalized, Magnetic Field Type, Rotating, Electro.n I' ../ . mechanical Energy Converter 170tt

4 Two-Phase Transformations and the Generalized Machine 254

';of +t 5 Fundamentals of System Dynamics 360

f" 6 Dynamics of Transducers 395

''l

7 Dynamics of Commutator Machines 421,~~l:, ) ~~

"",,) . OU1; ~i III .~1~:k}~~i 8 Dynamics of Induction Machines 477,.~~ "'I', ,r, (1 ~';, 1t1 .llj'I, .... ! 9 Dynamics of Synchronous Machines 508

'-:1:! ~.~')".1: -'

.~.~

);f 10 Generalized Analysis of the nm Winding Machine 545/

'~:;:'

\rf!~r"'! .j. 2; t Space Harmonic Analysis in Machines,Ii; 11 603 ~i ~. '~ Index 639

;,;;1 I:l '1'1 i

~. .~tt4 !:'.m ' ..

'JO"IIJ.b

Xy

~

"'-- -

pc H A T E R o N E

,

~

The Dynamic Equations of Motion

of Electromechanical Systems

,I.:~:;

1.0 Introduction

'{-; '. Electromechanical energy conversion is the result of electrouynamic interactions and a rigorous and thorough treatment requires a st1ldy of moving, charge-carrying material bodies in electromagnetic fields. However, for quasi-static (low-frequency) and low-velocity electromechanical systems the dynamic equations of motion can be formulated to a high degree of accuracy in terms of lumped electric circuit parameters that arerf. evaluated from static field solutions.'"

1 The subject of this book is the characterization of electromechanicalI systems by lumped parameters. This involves three major problems: (I)

a physical description of the system, (2) derivation of the differential equa,W tions of motion for the system, and (3) solution of the equations subject\i

to the specified operating conditions of interest. The physical description . ~UJ

1,;; of the system entails the establishment of an idealized model whose

character is determined by the physics of the problem and whose ,+ ~i:.',~ .. ~ completeness depends on the application. The equations of motion

\ .~'

J are obtained from the idealized model in several ways, some of which ,:; ~ are covered in the present chapter. The solution of the equations of

~ '-,'; /, motion is in general difficult because the equations for any energy confit version system are nonlinear. for some applications the equations can

I, Ii be linearized (e.g., for small signals); in others a change of variables

l~' J,,' 1 leads to simplification. In a few cases it is necessary to solve the * For a discussion of the relationship between circuits and fields for systems in ~......... relative motion see R. M. Fano and L. J. Chu, Fields, Enel'!:';'. alld Forces. John Wiley.

' .. '.. New York, 1959.

\ ~ t ...........j;:~.fIo... _~_ )- -_.

ELECTROMECHANICAL ENERGY CONVERSION 2

complete, nonlinear equations, and machine computation techniques may bc requi red. The types of nonlinearities encountered and the simplification possible depend upon the electromechanical systems and their applications.

1.1 The Various Approaches to the Study of the Dynamics of Electromechanical Systems

The dynamic equations of motion of electromechanical systems can be . determined from physical laws using either force density from electromagnetic field theory or an arbitrary displacement and conservation of energy to obtain the mechanical forces of electrical origin. Alternatively, the equations can be obtained from variational principles applied to selected energy functions. It is difficult to state if either of these approaches is more basic, particularly since both will lead to the correct results if properly applied. There is, however, a great difference between them as to the formality of the analytical techniques employed.

The application of force laws to obtain the equations of motion is probably the least formal analytically and requires a good deal of insight and judgment when dealing with complicated systems containing many variables. The electric coupling terms resulting from mechanical motion and the equations of motion for the electrical and mechanical parts of the system are obtained from known force laws, such as Faraday's law, Coulomb's law, Kirchhoff's law, and d'Alembert's principle. The method of arbitrary displacement and conservation of energy is then used to obtain the mechanical forces of interaction between systems, because it lends itself easily to the derivation of mechanical force equations expressed in terms of electric circuit quantities. Alternatively, these coupling terms can be obtained by integrating force densities obtained from electro

magnetic field theory. The derivation of the equations of motion from variational principles

is significantly different from the previous method. First, one establishes a common terminology for all types of systems, whether electrical, mechanical, thermal, acoustical, etc., by defining state functions (energy functions) in terms of sets of generalized variables. Then by the use of a single fundamental postulate, e.g., Hamilton's principle, the equations of motion for all systems including any coupling terms are determined. The variational approach is quite formal analytically; and, as a result, insight into physical processes can be lost in the mathematical procedures. However, if the method is properly understood, and if adequate attention is given to the selection of the state function, physical insight can be gained because of the generality of the method. The variational method is one

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 3

of the most pnwcrful techniq lies of dynamics; and, .tltllout!h beginners may tend to become "crank turners," the long-term value of variational techniques precludes their dismissal merely on the ground of conceptual difficulties or lack of physical insight when employed by the novice.

+

1.2 Fundamental Relationships in Electromechanics

j

The fundamental force relationship of statics is that for equilibrium the summation of all forces acting on a body is zero. This basic concept was used by d'Alembert who postulated that the sum of all forces equals zero for dynamic equilibrium of mechanical systems, * For a multiloop dynamic system d'Alembert's principle requires that at the kth mechanical nodet

2: r

(A, - fk) = 0 (1-10) I-I

,. ELECTROMECHANICAL ENERGY CONVERSION the force equation is expressed as follows: the sum of all voltage drops around a loop (kth loop) equals zero,

" ek, =0 (I-Ie)~

i--l

where ek, = the ith voltage in the kth loop. The continuity-of-charge relationship or continuity of current is

expressed by stating that the sum of all currents into a node (kth node) must equal zero. $'

1

Lr

ikl = 0 (1- Id) I-I

where iAj = the ith current Howing into the kth node.

D'Alembert's principle, Eq. I-la, the continuity-of-space relationship, Eg. I-Ih, and Kirchhoff's laws, Egs. I-Ie and d, express the complete equations of motion for electromechanical systems providing the mechanical forces of electrical origin are included in d'Alembert's principle, Eq. I-la, and the electric voltages and currents used in Kirchhotf's laws, Eqs. I-Ie and d, include the effects of mechanical motion.

The fundamental laws needed to study the dynamics of electron'~~hanical

devices from a circuit viewpoint are now complete. Unfortunately, the mechanical forces of electromechanical coupling, when expressed in a form easily used for electromagnetic field problems, are not readily adapted to an equivalent circuit treatment of connected electrical and mechanical systems. The direct extension of the macroscopic force equations in terrr.s of electromagnetic field quantities to force equations in terms of electric circuit quantities is difficult by integration methods unless the physical device is quite simple. It proves easier to derive these coupling terms, using an arbitrary displacement and the conservation of

energy.

1.1.1 Mechanical Forces of Electromechanical Coupling Derived by an Arbitrary Displacement and Conservation of Energy

~

The first step in analyzing a complicated electromechanical system by ail arbitrary displacement* and conservation of energy is to reduce the system containing electromechanical coupling terms to a minimum. To do this, separate out all purely electrical parts and all purely mechanical parts of

~' if,i,'",',,",

I, 'I the system including losses, as shown in Fig. 1-1. This separation procedure is carried out to the extent that each electrical terminal pair is For treatments of the principle of virtual displacement, see Goldstein, loc. cit., or E. T. Whittaker. Analytical Dynamics, Dover Publications, New York, 1944.

1

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSl EMS 5

coupled to one energy storage, either electrical or magnetic. Any internal interconnections between circuits that are coupkd to different energy storages are included in the external electrical network. The Illechani,cal variables represented by the mechanical terminal pairs arc those whi~h

affect energy storage in the ekctric and magnetic fields. Any purely mechanical couplings between mechanical variables such as gear trains, springs, etc.. are included in the external mechanical network.

Compte-t~ fiectromechamCOtI System . _ _ __ r------------------------. I

'.1_"--++ tt flectucal Network' (Ieetflcal ElectrICal

lQualllJlls of motton fromcnput 10 " Input toKirchhoff's lawssY5tem I coupling fields -

Electromechancal

L-t:

Network (repfesentlng coupling

fre!dc,-electrlC .Illd IT1.I~rlr.ll()

!:.ljUatlOfls 01 Fllohon from conSerVatIon 01 t'nt"I~Y and ,.HI arblltMy ulspl.lC\.'rlIent

(f,),+ '-

Mechanical Network: + t - I Mechanical f MechanicalEQuatlolis of moflon fromi:t Impul fo Input 10d'AleflltHHI's prillclple

system coupling fields! II and Conllnl,Jlty of ipace

ji I - I

I IL J

Fig, I-I. Simplification of electromechanical system for analysis by an arbitrary displacement and conservation of energy.t

1< (

The separation procedure results in the general conservative electromechanical coupling network depicted in Fig. 1-2 in which there are n electrical terminal pairs and m mechanical terminal pairs. By virtue of the lumped-parameter approach, each electrical terminal pair will be

il.' coupled to either a magnetic field energy storage or an electric field

iJ energy storage. To fix ideas, assume that the electrical terminal pairs I ~ i ~ I are coupled to electric field storage and terminal pairs, I + I ~ i ~ n are coupled to magnetic field storage. The total stored' energy W in the coupling network is given by

W = We + Wm " (1-2)

~ where W, is energy stored in electric fields and W m is energy stored il) magnetic fields.

t The lumped-parameter representation of the coupling network of Fig. 1-2 requires the system variables to be functionally related. For example,

, ..=eM.... . __, .

~

--

7 6

r .

ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS

--" ... +

("1

il; +-...:....-~---

Electrical Inplll~ to U. coupling

fields 0-=--- _

in

+ Un

fj ~

+ ~({')I

XI

fk -...::~----

0-:; ~--- ({.IkMechanical

mputs to X. coupling

fields 0-------

~

+ -- ({.Im %"1

,1

Fig. 1-2. Dcfinition of coupling system for and conservation of energy.

Electromechanical Network: (representing couplmg fields -electric and magnetic)

All dissipative elements removed to external circuits; therefore system is conservative.

.:tR g,~ ~~ r ~q

f10rJ ,,'

'1'''

, "'

1"

analysis by an arbitrary displacement

.~

if the electromcchanical coupling network can be characterized electrically lincar inductanccs. the kth nux linkage I~ givcn hy

by.

.:\k = " 2: i =,11 I

Lk;ii (1-30)

where tL.~ inductances L Ai Xb ,xm '

are functions of the mechanical coordinates

LAi == Lki(x" ... ,xm ) (1-3b)

Similarly, for a system of electrically linear capacitances there would occur

qk = 1

2: i,~ I

Cki/li (I-Jc)

Where the capacitances Cki Xl' .'" Xm ,

are functions of the mechanical coordinates.

Ckl = Cki(X" ... , X",) (1-3d)

In general. the system may be nonlinear; thus a set of parameters Land C cannot be defined. In such cases only general functional relationships among the variables can be established, such as

flux linkage and current of kth inductor: I~

or .:\k = ,\(i/+" ... , i,,; xI> ... , x",) ( 1-4a)

ik = ik("I+I, ... , '\,,; xI> ... , XIII) ( 1-4h)

voltage and charge of kth capacitor:,~!

" ,.!~

or Uk = Uk(qlo ... , ql; X Io .. , X",) (l-4c)

\:! ~ qk = qk(l'lo .. , VI; XIo , X",) ( 1-4d)

"",,'

r"J.

't~

;\.-""

~;

In any case, regardless of whether the relationships between the varia hies are linear or nonlinear, the relations are restricted to be single-valucd functions because it is assumed that the energy stored in the electromechanical coupling fields can be described by state functions.

The assumption that the stored energy in the electromechanical coupling network is a state function forces it to bc a single-valued function of the system variables. independent of the derivatives and integrals of the variables. Thus. the stored energy W may bc a function of the instantaneous configuf

8 ELECTROMECHANICAL ENERGY CONVERSION

function is a stronger one than the requirement that the expressions of Eqs. 1-4a-d be si ngle-valued functions. Stored energy that is a state function always yields singh~-valued internal constraints; hut single-valued inl\:rnal constraints alone do not guarantee that the stored energy is a state functillll (see Sec. 1.4.6).

In sUllllllary. the constraint of requiring the energy functions to be state functions places the following limits on the electromechanical system:

I. The lumped parameters evaluated from electromagnetic fields must be derivable from static (zero-order) fields. *

::. The functional relationships between variables (Eqs. 1-4a-d) must be single-valued.

3. Hysteresis cannot be included in expressing the functional dependence of the variables in Eqs. 1-4a-d. This does not mean that the loss due to hysteresis cannot be represented by a resistive element located outside the coupling sy.,tem.

Restricting the energy functions which describe the electromechanical coupling lields to b . , An; i/+1> . , i,,; x" ... , xm;fJ, .. ,fm) (1-5a) j

l W = iq'q,:AJ! 1.,\n;Xtxm '!. '!! 1 ,L . L [v;(ql"'" o, ... ;0;0 0;0... 0 i~ I j~l " ql; Xl' , " Xm) dq, ..t.~.

..~~ .,

+ i;(>\;r\> , A;,; xi, ... , x;,,) dA; ~ ~i~ + f;(q;, , q;; 1.;+10' .. , A;,; X;, . .. , x;") dXiJ (1-5b)

where primes denote variables of integration. The energy function of 'jr

tj Eqs. I-Sa and h is evaluat!:d by performing the line integral over all the ~

system variables. The constraint of the proper single-valuedness of the

, functional relationship indicated in Eq. J-5h to yield a state function allows the evaluation of the line integral by choosing any convenient path of j

I.;.."i ;~/

integration. Also, because of the interdependencies of the variables in

Eq. 1-5b. it is clear that of the six variables 'Ii' Vi' ii' Ai' fj, and xj there

See Fano and Chu, loc. cit.

-

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS . 9

can be only three indepcndent variables, the other lhn:!: variah1l.:s heing required to satisfy the interna~constraint equations of the form

Vi - 11;(lJ" ... ,q,; Xb ... , x",) (1-611)

il = i,(A/1 1, , An; Xh ... X,.,) ( I -flh)

jj = jj(qh ... , 'II; A'+h' , An; Xh ... , X",) ( 1-6(')

In view of these constraint equations, it follows that the energy function W can be expressed as a function of three independent variables. which lor the constraint equations I-oa-(' are 'Ii' ,'1/, and x j yielding :In energy function

W = W(ql"'" ql; 1.'+10 An; Xh ... ,Xm ) (1-6d)

Even though qi' \, and x j have been considered as the independent variables above, the single-valued relations of Eqs. l-6a-c can be solved to allow any three of the six variables to be treated as independent.

In gener,l!, in electromechanical systems it is possible to evaluate constraint equations 1-6a and h by solving for the electric and magnetic static field solutions determined by the physical configuration. The evaluation of the constraint equation for the force fj is usually much more difficult. Thus it is common practice to find the constraint relationships for the electrical variables and to use these to evaluate the stored energy, utilizin'g the fact that since the energy is a state function the correct energy function can be obtained by keeping all electrical variables Ai and qi zero when assembling the mechanical system and then holding all the displacements x j constant when establishing the final values of the electrical variables. For this path of assembly of the system the energy function becomes

W(qh ... ,ql; 1.1+ 10 An; x .. ... xm)

i qlt .. lq,; ),,+l ... An;xll ...xm n = L [v;(q~, ... , q;; Xh x ) dq;m0.....0;0.....0;,l'1.......... /_1 + i;(A;+1... , A~; XI> x ) d,\;J (1-7)m

The energy functions given by Eq. 1-7 can then be used to find the force constraints fj and hence the force of electromechanical coupling.

The forms of the stored energies and the nature of the network in Fig. 1-2 indicate that n electrical variables and m mechanical variables can be specified independently, This can be interpreted as meaning that the coupling network has (n + m) degrees of freedom. With the usual interpretation that a degree of freedom represents one independent energy storage element in a lossless system, the coupling network appears to

10 ELECTROMECHANICAL ENERGY CONVERSION

have (n + m) independent energy storages in spite of the fact that in Eq. 1-7 energy storage was assigned only to each of the n electrical terminal pairs. This apparent ambiguity is resolved by realizing that mechanical energy supplied at the mechanical terminals goes into either electric or magnetic field storage. Consequently, in a coupled electromechanical network the electric and magnetic fields can be interpreted as storing both electrical and mechanical energy and hence can represent both electrical and mechanical degrees of freedom.

With the electromechanical coupling network of Fig. 1-2 specified, the terminal characteristics of the network can be found. According to the discussion above, one variable at each of the (n + m) terminal pairs can be specified independently. First, consider the I electrical terminal pairs that are coupled to electric field storage. When the qj and Xj are specifie. independently, the current in the ith terminal pair is t

. dqjI = (l-8)

I dt

and the Yoltage Vi at the ith terminal pair is given by the internal constraint of Eq. 1-6a. Next, consider the (n - I) electrical terminal pairs that are coupled to magnetic field storage. When the Aj and xj are specified independently, the voltage at the ith terminal pair is given by Faraday's law

d\ V, = di (1-9)

and the current il is given by the internal constraint of Eq. 1-6b. Thus, the Yolt-ampere characteristics at the electrical terminals have been determined so the effects of the electrical part of the coupling network can be ineluded in the equations for the external electrical network described in Fig. 1-1. It should be mentioned that instead of specifying qj for the I electric field storages and A; for the (n - I) magnetic field storages, the voltage Vi at the electric field storages and i j at the magnetic ficld storages could have been considered as independent, in which case the internal constraints of Eqs. 1-6a and b would be used to obtain qj and AI for inclusion in Eqs. 1-8 and 1-9.

The next problem is to find the force due to the electromechanical coupling. Since the m mechanical terminal pairs are characterized by m independent variables, it is possible to consider each mechanical terJl1inal pair individually to find the electromechanical force. Defining the force (le)k shown in in Fig. 1-2 as the force applied to the kth mechanical coordinate (node) by the electromechanical coupling network, the force (le)k can be found by considering that an arbitrary displacement

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS II

dXk occurs in the kth mechanical coordinate in time dt with all other mechanical coordinates held fixed, i.e..

dXJ = 0 for j 'I- k

The electrical variables change during the time ill. and these changes are completely arbitrary except that the internal con~traints of Eqs, 1-6a and b must be satisfied. This means that only one electrical variable at each electrical terminal pair can be changed arbitrarily. During the arbitrary displacement the conservation of energy must hold. The various energies involved in the arbitrary displacement are:

energy supplied at electrical terminals = n2: v,i; dt

1=1 (I-lOa)

~ .. ;

energy supplied at mechanical terminals = - (I.h-':k dt = - (le)k dXk (I-lOb)

change in stored electrical and magnetic

energy of coupling field = dW (I-JOe)

where W is the total stored electrical and magnetic energy of the coupling fields (Eq. 1-5b)

energy lost in dissipation = 0 (I-IOd)

i

f.!,

(1-11)-(i')k dXk + i V,il dt = dW ;=1

All lossy elements are removed from the network of Fig. 1-2. The conservation of energy requires that the sum of the input energy

must equal the change in stored energy, thus, from Eqs. 1-10: j

.,it.

From this expression the force applied to the kth mechanical node by the electromechanical coupling field is:

(f,)k = d.i.... ( i viii cit X k ;=1

- dW) ( 1-12)

When the n independent electrical variables and the fit independent mechanical coordinates are specified, Eq. 1-12 gives the force applied to the kth mechanical node and the velocity of the kth mechanical node is dxk/dt = Xk' Thus the force-velocity characteristics at the mechanical terminal pairs of the electromechanical network of Fig. 1-2 have been specified. The force of Eq. 1-12 can be included with d'Alemhert's

it

.. I'

12 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF ELECTROMECHANICAL SYS rEMS 13

principle to write the equilibrium equation for the kth mechanical node aN energy estahlishes that the energy input from all sources j, 'l(lrct! as magnetic field energy W",(A" ... , An; XI> .. , X n,).

ilk - Uk + Umh + (j,)d = 0 (1-13) Wm = input electrical energy + input mechanical energy (1-14a)

where Pk is the inertia force, fk the externally applied mechanical force,

and (t;"h the force applied hy mechanical springs. All three of the or

terms arc considered to be included in the mechanical network of Fig. 1-l.

J. A, .....Ann

W",(AI>' .. A,,; x., ... , x",) = .~ i;(A;, . .. A:,; x;, . .. ," ~~ , '.' I.'Eqs. 1-6a and h. Consequently, this force is the true force regardless of T .,,:- f/Al, ... , A", XI' .. , x m ) d'\j (1-14b)0 .....0 J .. I

t!le external lerminal colIslraints that may he imposed on the coupling

This is tonetwork hy the external electrical and mechanical system. for the general case of

emphasize the point that only the inlernal constraints had to be satisfied II A graphical plot of the total energy Wm'

A;(x;, ... x;"; i;, ... , i;'), where A; is a nonlinear. single-valued function in the arbitrary displacement.

The mechanical force (f,h of electromechanical coupling contains II

terms due to two types of energy storage fields, electrical (W,) and

magnetic (Wm ). At low electrical frequencies and low. mechanical .

velocities such that the electrical system can be represented by lumped:'" .~

parameters, all electric energy storage (W,) will be in capacitances and.

all magnetic energy storage (Wm ) will be in inductances. Thus, ,he two: ~rlca,

J sy~tem

problems of electric and magnetic field coupling can be treated separately

---

and the results can be combined, with the proper external or terminal

constraints, to describe a system having both electric and magnetic field

coupling with a mechanical system. The forces caused by each of these

two coupling fields will be treated in the next two sections.f..

1.2.2 Mechanical Force Due to MagnetiC Field Coupling.J:'\'. '':.'A system of current-carrying coils is represented schematically in

Fig. 1-30. The only important, or first-order, electromagnetic coupling

field between various elements of this network for quasi statics' and for f'

To evaluate thelow-velocity mechanical motion is the magnetic field.

stored energy in the magnetic field, assume that the final flux linkage AI

Fig. 1-3a. Coupled current-carrying coils.

in each of the coils in Fig. 1-30 and the final positions of the coils Xl are

For any givenobtained by any arbitrary paths compatible with the internal constrain

ts of the displacements and currents, is shown in Fig. l-3b.

( of both the electrical and mechanical variables between zero and their system, where A; is a single-valued function of the x j and ii. the total

final values. Assuming all mechanical storage elements and all non stored magnetic energy Wm is determined uniquely by the paramete

rs

coupled electrical storage clements plus all dissipative elements to be XI>' .. , x", and AI,' .. ,A" and only depends upon the final values of these

The total stored magnetic energy is a state function. Theif ;1 external to the electromechanical coupling field, the conservation of J parameters.~' amount of the storeL! magnetic energy which is supplied by electrical,:,f:}:~

.

For alternate treatments see E. Fitzgerald and C. Kingsley, Electric Mach

ines, sources and that which is supplied by mechanical sources depend upo

n McGraw-Hili. New York, 1952; R. E. Doherty and R. H. Park, "Mech

anical Force how the system is assembled in reaching its final state. The energ

y j- between Electric Circuits," Trans. AlEE, Vol. 45, 1926, pp. 240-252.

...~)

.1.....,...;;

j...I.."

~.

14 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 15

'I

i-z ii

(Stored magnetic energy~i"" i~d)"~

, 0

~~~~

Magnetic coenergy.

I i" A'di'o n A

i~t 71

Fig. I-lb. Path of operation to reach final energy when mechanical coordinates are held at xi = XI' xi = x 2 , , X~ = x m while the electrical variables are simultaneously brought to their final values.

",I

&1 ij

X2(2:i..:c,;;;i{. .i~)

i2 ii

-.. ,. f.

ill i~

Fig. I-le. Path of operation to reach final energy when mechanical coordinates are held at xi = XI' xi = xi, x~ = x;. while the electrical linkages >'1 are brought to their final values, then the mechanical coordinates are brought to their final values.

j

supplied from ekctrical sources and mechanical sources 1',)1' a particular path of asst:J1lbly or Ihc system shown in Fig. 1-.111 i, drawn in hg. I-.~c. From these ligures it should be clear thal it is possible to arbltral'lly supply any portion of the total energy from either source by the prllper choice of path used to assemble the system. For example, hold all !lux linkages at zero and mechanically assemble the system, and then establish the flux linkages with the mechanical coordinates held at their final values. The energy stored in the magnetic field for the given values of flux linkage will have LV be supplied by electrical sources. For this case, the stored magnetic energy is

Wm(AI> ... , An; XI . , xm)

"I .... '''" n=2: i;(/,;, ... , A~; XI, ... , x ) dA; (1-15) mJ0, ....0 ,-I 1 where Wm is evaluated as the integral of idA for any fixed spacing, i.e.,

,~! all Xl are constants.

..1t2 EXAMPLE lEI

As an example of the method of evaluation of the integral of Eq. 1-14b. consider the case in which

n=3 '$' " and the system is electrically linear '1

il = TllAl + T 12AZ + T13A3 i2 = TZ1A I + Tn)o.z + T 23 A3 (lEI-I)

-"I i3 = T 31 A1 + TnAz + T33 A3 ~'" /. ~ \1~)with the T's functions of the xj only and

T IZ = T2b T13 = r3l> T Z3 = Tn f)J,

The meaning of Eq. 1-14b :1

J"I'''2'''3 3 I.'

Wm = 2: i;(A;, A;, Ai; Xl> , X ) dA; (IEI-2)m0,0.0 i=1

is that with the xj held constant each flux linkage is brought to its final value holding all other flux linkages fixed. The order in which the flux '" linkages are brought to their final values is immaterial because the energy is a state function. For the purposes of illustration assume that the flux

--~

16 ELECTROMECHANICAL ENERGY CONVERSION

linkages are brought to their final values in the sequence AI> .12, .13 ; then Eq. IEI-2 can be written out as

rA1 0 0 0fA' l'A2'Wm = Jl 1;(,\;,0,0) dA; + I~(AJ> .1;.0) dA; 0,0.0 Al'O,O

JAPAz.A l+ ii(AI> Az, A;) dA; (IEI-3) A1.AZ.0 . Wm = t l TIlA; dA; + f: 2 (T21 A1 + TZ2A;) dA;

+ foAl (T3 \A 1 + T 32A2 + Tn'\;) dA; (11-4)

Recognizing that unprimed variables in the above integrations are held fixed, there results

W m = !TllA~ + r 21 A1A2 + !f'22A~+ 1'31 A1A3 + T32A2A3 + tr33Ai which can be written as

J 3

W m, = L L !TijA/A] /~I j~ I

which is the conventional expression for energy storage in a linear magnetic field device.

" The interchange between electrical and mechanical energy via the stored energy in the magnetic field is a direct manifestation of the energy con version process. The fact that the stored energy can be determined for any configuration of the system, and that this stored energy is a state function defined solely by the functional relationships between variables" and by the final values of these variables, provides a powerful tool for determining the coupling forces of electromechanics.

Now that the stored magnetic energy has been determined, the arbitrary displacement and conservation of energy as expressed by Eg. 1-11 can be used to evaluate the mechanical forces on the system of Fig. 1-3. Assuming that the stored electric field energy W~ is zero, it follows from Eq. 1-12 that

" n

C('.). dXk = 2: II dA, - dWm (1-16) /=1

where dA; = v/ dt. To obtain Eq. 1-12 and hence Eq. 1-16, an arbitrary displacement dx, of the kth mechanical node was assumed to take place. At the samc time no explicit restrictions were placed on the changes in Aj and I ; consequently, it may appear that the force (J;)k will depend on

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 17

how).., and i; vary during the arbitrary displacement. It will be shown, however, that th~/ force (.f..}k is independent 0/ the mriatiollS (If ,\, lind i; tlurinK the arhitl'w')' displacel1lent pl'Ol'ided the changcs in '\; al/l/ i, ji,f!oll' the /unctional relationships (the internal constraints) oj' thc I.I'\ICIII hcc Fig. 1-3b and Eg. 1-7). In other words, the fact that only one of the two variables A; and ii can be treated as indepcndent must be recogni1SJ as an internal system constraint.

The arbitrary displacement is taken at the point defined (see Fig, 1-3b) by

Ai = >';(11 ... ill; xI> ... , X",) (1-l7a)

This expression establishes the dependence of Aon i and X which must be maintained during the arbitrary displacement defined in obtaining Eq. 1-16. This single-valued dependence results from the stored magnetic j

,energy Wm= ~f/",(il> ... , in; XI> . X",) (1-17b)

being a state function. The second term on the right of Eq. 1-16, dWm, is a total differential (see Eq. 1- 11) and becomes

- 8Wmd ~: 8W"'d'dWm - -

18 ELECTROMECHANICAL ENERGY CONVERSION EQUA"I,JNS OF MOTION OF ELECTROMECHANICAL SYSTEMS 19

Substituting Eqs. 1-18 and 1-21 into Eq. 1-16 yields It is possible to put Eq. 1-27 in a simpler form by writing it in terms of the magnetic coenergy W:n which is defi ned as:

- - 0Wmd ~, . all; d (/.),k dX" - -')-- x" + L. I; ~ x" , l'l .... ,/n n.. ,., 0'of(;"" 1=1 uX" W m = .~ A;(lp ... , I,,; xl> ... , Xm ) dl, ( 1-28)

~.(~fJA;d) ~fJWmd'+ .L. I, L. Y I, -.L. y- II (1-22) 1 __:1 r=l I, Iz~1 I,

Rearranging terms and interchanging indices in the last term change Eq. 1-22 to the form

- oW " aA') " (- oW "aA)(Ie)" dx" = ( a m + ,2 i; ~ dx" +2: ~ + 2: i, ~ di;x" I-I uX" 1=1 uI; ,=1 ull ( 1-23)

In order that the force (f,)" be independent of the change in i; and AI during the arbitrary displacement, the coefficient of dil in Eq. 1-23 must

be zero. - oWm ~. OA, _ 0-,,-.- + L I, -:;;:- - (1-24)

u1i r= luI;

It can be seen graphically how this condition makes (Ie)" independent of the changes in i j and II; by referring to Fig. 1-4. For any di; there is a corresponding dll; (two examples are shown in Fig. 1-4); thus, if (I,)" is independent of di j , it is also independent of dA;.

To show that Eq. 1-24 always holds, the definition of stored magnetic energy given by Eq. 1-15 will be used and integrated by parts (f idA = iA - J Adi) to obtain the alternative definition:

n nlill.... i n Wm = ~ i,A, - ~ A;U;, ... i~; xl>' .. , x m) di; (1-25)

,-1 0... ,0 , 1

Substitution of Eq. 1-25 into 1-24 and evaluation of the indicated derivatives (keeping in mind the functional relationships of Eq. 1-17a)

yield

- () Wm ~. oA, ~ . oA, \ \ ~. OA, 0 ( 26)-.,.- + L. 1,-:;;:- = - L. I,~ - "; + II; + L l,~ = I01; r-I uti r= I uti r= 1 uli

which is always satisfied. Consequently, the force (Ie)" is always given by:

(/.) = - oWm(i1> ... , in; XI> , Xm) " ax"

+ i i; oA;(;1> ... , in; xl> .. , X m) (1-27) ;-1 aX"

This is a perfectly general expression for (I,h which holds regardless of how A; and i; are changing with time in the system.

0.. ,,0 .-1

~: di di ~r 'T 'i_

~j f- I i dX; I

dX, Change dUring K"1 arbitrary displacement

Ai (i;, ,i~iXl',,,,XRt,,,xm)

I Change dunng arbitrary displacement I

r

I I I I I

;"_~,1i\jt..~.f

,-,. (," '''i' ,xk +dxk ... .x )'\ I' , n' m I I

I t o ii i , ~

Fig. 1-4. Illustrating how Ai and ii can change during an arbitrary displacement dXk.

The relation between energy and coenergy has already been established as a consequence of writing Wm as in Eq. 1-25 thus:

n Wm = L i;A; - W,:, ( 1-29)

;=1

This relation is illustrated graphically for the ith CircUit In Fig. 1-5. Substitu';')n of Eq. 1-29 into Eq. 1-27 and subsequent simplification lead to

aw'(' .(I.)" = m I" , In; XI> . , xm ) (1-30) aXk

If:..o..L

20 ELECTROMECHANICAL ENERGY CONVERSION

Equations 1-27 and 1-30 give the force U.)k when the displacements and currents are used as the independent variables. If it is desired to express the energy with the flux linkages ('\) and displacements (x) as independent variables, Eqs. 1-27 and 1-30 must be modifted. For this new functional dependence the stored magnetic energy is expressed as

Wm = Wm(>'lo .. , , A,,; xI> ... , xm) (1-31)

and i j = ij(AI> ... , A,,; xl> .. , x m) (1-32)

AI

AI

c -,Iiu

Fig. 1-5. Graphical relation between magnetic energy and coenergy.

Since \ and Xj are the independent variables the differentiations of

Eq. 1-16 yield:

- aWm(AJ, ... , An; Xl> .. , x m ) d AU;')k dXk = Xk OXk

~ aWm(A" ... , An; Xl>"" X m ) dA/ i-I OA;

n + L i;(AI> ... , An; Xl> , X m ) dA/ (1-33)

;=,

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 21

With the stored magnetic energy given by Eq. 1-15, the last two terms of Eq. 1-33 subtract to zero, giving for the electromechanical coupling force U,oh applied to the kth mechanical node:

(n = - ()Wm(,\\ 0 An; XI> ... xm ) (1-34), , , I. OXk

This force can also bc evaluated in terms of the coenergy When Eq. 1-29 is substituted into Eq. 1-34. remembering that the'" and ,Ij arc the independent variables. there results:

aw,;,p'l> . An; Xl> . , , X",)U.h = OXk

_ )- A, ai;(Al> . , . An; Xl> .. xm ) (1-35) ;::;'1 ' aXk

Equation 1-35 is an equivalent way of expressing Eq. 1-34. The several forms of the electromechanical coupling force ef..)k applied

to the kth mechanical node by a magnetic coupling field as found by an

TABLE I-I. Mechanical Force Caused by Magnetic Coupling Field

= t,,A. .-1i i; d)..;Stored magnetic energy W m (1-15) 0"".0 ro Magnetic coenergy W~ = .,,';. ~ A; di; (1-28) 0...0 I-I

Relation between energy and coenergy W.. + W~ = L i,A, (1-29)

'-I

Conservation of energy during arbi L i, dA, = dW.. + (f.)t dXt (1-16)trary displacement dXt (Iossless) 1-1

Independent Force Evaluated Force Evaluated Variables from Stored Energy from Coenergy

Currents i, ([.) _ -oW.. i. OA I ([.) = iJW~t - -- + 1/Coordinates x, } iJXk I-I OXt k OXk Flux linkages A, 1 ([.) = -oWm (f.)k = 0 W,~ _ I AI .oil Coordinates x J J k OXk OXk I_I OXk

arbitrary displacement of the kth mechanical node XI. are summarized in Table 1-1. The four expressions for U.)k given in Table I-I are equivalent and will yield identically the same force, which is the true force, for a given

22 .

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 23ELECTROMECHANICAL ENERGY CONVERSION

state of the system, i.e., for a given set of ii' Ai> and Xj' In order to find the dynamic path of the system the force of Table (-I must be used with d"Aklllbl:rt's principle and Kirchhoff's laws to establish the cquations of dynamic equilibrium.

The results of Table I-I are complctely general and independent of electrical source variations (assuming low electrical frequcncies and low mechanical velocities such that a quasi-static solution is valid). It is worthwhile to examine some of the results more closely.

For instance, the force obtained from the coenergy with i l and x) as independent coordinates was given by Eq. 1-30

(f.)k = oW~(il> ... , in; Xl> ... , xm) (1-30)oXk

It has already been shown that this force is independent of the changes in A; and ii which take place during the arbitrary displacement; consequently, this expression is valid regardless of how>"; and ii vary, if the variation is compatible with the internal constraints, and therefore it is a general expression for the force. On the other hand, considering Eq. 1-30 from a mathematical point of view, since il and XI are independent variables the partial derivative is taken with respect to Xk, holding all other x's and all i's constant. The holding of the i's constant is a mathematical restriction imposed by the selection of independent coordinates and has nothing to do with electrical terminal constraints. The mathematical restrictions are often misinterpreted as electrical terminal constraints, and some confusion about the generality of the force expressions results.

Statements similar to those just given about Eq. 1-30 can be made about all the force equations in Table 1-1. These force equations are general; the mathematical restrictions placed on the derivatives by the choice of independent coordinates have nothing to do with electrical terminal constraints in general.

On the other hand, the general expressions of Table 1-1 can be used to interpret specialized electrical terminal constraints. For instance, if all changes in flux linkages d>"; are constrained to zero, there can be no energy flow between electrical sources and magnetic fields; conseq uently, energy conversion must take place solely between the magnetic field and the mechanical system. This is illustrated by noting that the electromechanical coupling force, when evaluated from stored magnetic energy with>"; and x) as the independent variables, is simply the negative rate of change of stored magnetic energy with respect to mechanical displacement with the flux linkages held constant. In this case the elcctrical tcrminal constraints of the special case coincide with the mathematical restrictions of the general case.

Another interpretation of this type can bc made by considering a system excited by electrical constant-current sources and by inquiring about the energy conversillll. From Tahle I-I the force is evaluatn! from the coenergy lIsing i, and x, as independent coordinates; the force is mathematically given by the ratc of change of magnetic coenergy with the currents held constant. In this case the mathematical restrictions in the general case coincide with the electrical terminal constraints in the special case. This leads to an interpretation of the coenergy W;n as a measure of the converti bility of electrical energy from constant-current sources.

When a system is electrically linear some general statelT1l:nts can be made ahnut energy conversion and about the relation between energy and coenergy. By electrically linear it is meant that the flux linkages are linear functions of the currents, thus

where

"

A; = n

L [i,i; (1-36) r= I

[I, =: [1,(Xh .. Xn,) = I" (1-37)

is a general single-valued function of the displacements. The use of Eq. 1-36 in the definition of Wm (Eq. 1-15) yields for the stored magnetic energy

;l .... IO n (n )WIN = L i; L Ii' di; z:

n

L n

-tli,i;i, (1-38)J0 .....0 ;=1 r=1 i-=d r=,-" J From Eq. (I -29) the magnetic coenergy is

n

W~ = L i;Ai - WIN (1-39) i=1

Substitution of Eqs. 1-36 and 1-38 into Eq. 1-39 yields the result:

fI n

W;" = Wm = L L 1!,J,i, ( 1-40) ;=1 r=1

Thus in the electrically linear case the stored magnetic energy is equal to the magnetic coenergy. This can be seen geometrically by considering Fig. 1-5 with a linear relation between A; and i;. "

The fact that the energy and coenergy are equal in the electrically linear case has led to the use of the two state functions interchangeably. Investigation of Table 1-1 shows that energy and coenergy must be distinguished; otherwise in the electrically linear case the sign of the mechanical force will be in error if the wrong state function is used.

24 ELECTROMECHANICAL ENERGY CONVERSION

Equation 1-16 can be used to write the conservation of energy for an arbitrary displacem'~nt c/.>;J.. as:

" 2: i; d>.; = dWm + U"h dXk (1-41) i ,

' __ ....... -.J'--.,--' ~

cl,,'rtrli.:.a1 Morcd Ill",,,; 1\;ll11Gl I Iliput liclt.1 ouqHlt

l'lll"rM)' energ)' Cliel ~y

Next, the individual terms on the right-hand side of Eq. 1-41 can be cV.lluated, assuming an electrically linear system. Using the force from Table I-I and the coenergy with i j and x j as independent variables, the energy converted from electrical to mechanical form in an electrically linear system is

" " 1 of. (1-42)U,)k dXk = ;~I '~I 2" o_~: iii, dXk

The change in stored field energy is found from Eq. 1-40 as:

_ ~. ,. .. ~. ~~ ~ 01;, ..dWm - ... ~ 1;,l j dl, + ... ~ '). Ijl, dXk (1-43)

; " , ; - - I , I - (IXk

According to Eq. 1-41 the sum of Eqs. 1-42 and 1-43 equals the electrical input power. When all the electrical sources are constrained to be constant-current sources,

di, = 0

and the electrical energy converted to mechanical form becomes equal to the change in stored magnetic energy. Thus, when an electrically linear system is excited by constant-current sources, the electrical input energy is divided equally between stored field energy and converted energy.

1.2.3 Mechanical Force Due to Electric Field Coupling-

The mechanical forces produced by magnetic coupling fields in an electromechanical system have been determined. A similar development can be made for finding mechanical forces due to electric field coupling in an electromechanical system. Consider the case where the only sign iticant stored energy is electric flcld energy (sec Fig. l-6a). The electrical stored energy W, can be found in a manner like that used to find the magnetic stored energy in the previous section. Assume that all purely electrical or mechanical energy storage elements plus all dissipative elements have been removed from the system (see Sec. 1.2.1); then, from

For an alternate treatment SL"C Fitzgerald and Kingsley, [(/('. cit.

.,

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS .25

the conservation of ener.gy the stored electrical energy W" must t:qu,il the input energy from all sources,

W, = input electrical energy + input mechanical energy (1-.44)

l"' .. ,,' ' W.(ql> ... , q,; XI' ... , xm ) =2: I';(q;, _. _, 'I;; x;, ... , X;,,) !/l/ n.....o ,. I l

'!'\'m,~ " I. ,I.'I I+ . ..:. Ij(q" - . - , q" XI' ... \",) d\j (1-45) n... :0 j I .

" ... ;;:.

f.".i"l

., ~

Fig. l-'a. C41ed charge-earrying conductors.

The expression for stored elee.trical energy as given by Eq. 1-45 is plotted in Fig. l-6b and c. The amount of the total stored electrical energy supplied by electrical sources and that supplied by mechanical sources arc dependent upon the manner in which the system is ,assemhled; however, since the electrical stored energy is a statc function, YV. can be expressed in the simple form:

W,(ql> ... ql; Xl> . , x )m

1"\ .....'11 I = 2: v;(q;, .. " q;; x", .. XIII) tlq; (1-46) n....n j I

26 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 27

. . jq,II; t Stored electncal energy = (} 1/; dq; qj f,\,

:w I:LEC IKOMECIIANICAL ENERGY CONVERSION

Now Eqs. 1-4l\ and \-49 lIsed with Eq. \-47 give the force U;.h as:

- i'IV ' ?q,) , (- i)W ' (1q)

U~). d.\. = ( -----;-,-,-" + ~ P, ~ tis. + ~ -i:!'+ 2 /', -i:!--~ tlv, (I-50)e\. I~I (','. j-I Vi '-1 Vj It can be shown quite readily that

- oW, !.; oq, 0 (I-51).-.-,- + .:... V, 0:;- = (,Vi ,-1 vV;

Therefore the forcc (f,). is independent of how Vi and q, arc externally constrained (providing Eg. 1-48 is satisfied) during the arbitrary displacement oecause the coef1icient of tlv, is zero in Eg. \-50. The resulting expression for the force is:

(f..) = -OW,,(PI""; 1',: x, ... , x",) k cJx.

~ V oll'(I'. I' . X X )+ ') "',1'-1,",," (1-52)I ':-1 OX.I

It is possible to put Eq. I-52 in a simpler form by writing it in terms of the electrical coenergy W;.. which is defmcd as:

ul ..... , ' W~ =

v

2 q;dv; (I-53)i0.... ,0 j.=, The relation between the electrical energy and coenergy is (see Fig. 1-6b)

, W, + W~ = L vjql (1-54)

'-I Substitution of Eq. 1-54 into Eq. I-52, making certain to keep Vj and xJ as the independent variables, leads to

(/')1( = oW~(VI< ... , v,; XI< , , x",) (l-55) cXk

Equations I-52 and I-55 give the force (f,). when the displacements xJ and the voltages /'i are independent variables.

Expressions can be obtained for the force (fe)' that are equivalent to.. Eqs. I-52 and I-55 but expressed in terms of the displacements X j and the charges qj as independent variables. Assume that the voltages are

expressible as:

VI = V;(ql' ... ,q,; XI> .. x",) (I-56)

Then the stored electrical energy can be written as:

W, = W~(qh"" q,; Xh , x m) (I-57)

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 29

The use of Eqs. I-56 and I-57 with Eq. 1-47 yields the result

(f.) = _ oWr(q" .. , q,: Xl, ' ,X",) ( I-58) ,. OXk

which is the correct force regardless of how Vi and qj vary during the arbitrary displacement.

Substituting from Eq. ]-54 for W, into Eq. I-58 and recognizing that the x j and lfi are the independent variables yield the force in terms of the electrical coenergy as

()W~(ql< . , .. q,: Xl' .. Xm)Cf.h = (lx. ~ Ol';(q,. ' .. ,q,: XI< .. Xm) - .;.. qi " (I-59)

i-.I ex.

The various forms of the force U,)' which have been found for electric field coupling are summarized in Table 1-2. The four expressions for

_'J' TABLE 1-2. Mechanical Force Caused by Electric Field Coupling

I'I" .. '" ~ 'd'Stored electrical energy W, = .. V, q, (1-46) 0.....0 1-1 Iv' .....r l "d'Electrical coenergy W' = L q, v, (1-53) , 0.....0 '-1

r.t ',t . ";,:~ , I

Relation between energy and coenergy W, + W; = L v,q, (I-54)'-I ,

Conservation of energy during arbi 2: VI dq, = dW, + ([.)_ dx_ ( \-47) trary displacement dx_ (loss less) ;~ '-I

.;1'; '\I~

Independent Force Evaluated from Force Evaluated Variables Stored Energy from Coenergy

Voltages v, } , -oW' oW:(f.h = --' + ') v oq, (1.)_ = oXCoordinates XI ax_ I~ I ox_ 1 Charges q, } -oW aw' 'ou,(1.)_ = --' (f.) , - L q-Coordinates XI ax_ , k = eXt I_I I ox_

~;

force (f,h are equivalent and will yield exactly the same force, which is the true force, for a given state of the system, i.e .. for a given set of qj,

i Ilj. and Xj' In order to find the dynamic behavior of a system the force of

of Table 1-2 must be used with d'Alembert's principle and Kirchhoff's laws to establish the equations of dynamic equilibrium.

38 ELECTROMECHANICAL ENERGY CONVERSION

The force expressions of Table 1-2 ~re;i completely general and the mathematical restrictions imposed by the differentiations indicated must not be confused with external electrical constraints imposed on the system. Equation I-58, which yields the force for any general case, shows that in the special case of a terminal constraint of constant charge, the energy is converted between stored electrical energy and mechanical energy with no input from the electrical source. The general expression for force of Eq. I-55 also shows that when a special external electrical constraint of a constant-voltage source is applied, the energy converted from electrical to mechanical form is equal to the change in electrical coenergy. The electrical coepergy can be considered as a measure of the convertibility of energy from constant-voltage sources through electric field coupling, In addition, it can be shown that when an electrically linear electric field system is excited by constant-voltage sources, the electrical input energy divides equally, half going to electric field storage and the other half to mechanical energy. These results are clearly analogous to those derived in the previous section for magnetic field coupling. They also indicate that for electric field coupling a constant-voltage constraint and a constant charge constraint are analogous respectively to a constant-current constraint and a constant-flux linkage constraint for magnetic field coupling.

1.3 Hamilton's Principle-

In the previous sections the equations of motion of electromechanical systems were obtained from force laws, These force relations were determined from basic experiments or postulates of physics, and they form a set of "differential principles," i.e., principles concerned with incremental changes in the system.

It is also possible to develop an alternate approach to the problem of describing the path of a dynamic system by postulating that the dynamic path of the system is determined by finding the extremum of certain integral functions. This alternate approach is called a variational method and is based on a set of "integral principles," i.e., principles relating to gross motion of the system.

When a system is completely described by one of the principles, the other principle can be derived. This is just another way of saying that one physical system is described by one dynamic path of operation regardless of how the equations of motion are derived. The integral principle

For a derivation of Hamilton's principle from the principle of virtual work, see Goldstein, loc. elf. or Whittaker, loc. cit.

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS jJ

usually considered to be the most fundamental is Hamilton's principle. Hamilton's principle can be derived for mechanical systcms from d'Alembert's principle and the principle of virtual work. However. Hamilton's principle proves to be significant for other than just mechanical systems, and as such overshadows d'Alembert's principle and is at least a more general relationship if not a more fundamental one.

The usual statement of Hamilton's principle is that the variation of the time integral of the Lagrangian L between fixed end points qi(tt) and

'''").~

32 ELECTROMECHANICAL ENERGY CONVERSION

representation of the meaning of an extremum is shown in two dimensions in Fig. 1-7. ,>,

The condition for an extremum of the function 1 is that the variation of the function I equal zero,

M = S f.1 2L(ql(/), ... , qN(/); (Ml), " . ,QN(I); I) dt .. 0 (1-61) 'I

subject to the end conditions that.

oqj(t\) = 0 and oqj(l2) = a for i = 1, 2, 3, ... , N

In the above equation the symbol 8 means the time-independent variation as used in the calculus of variations and is analogous to a differential in ordinary ditTen.:ntial calculus. (Sec Sec. 1.3.2.)

Equation 1-61 is Hamilton's principle. It is now necessary to see what set of relationships must exist among the N coordinates, the qj, and their N derivatives, the q" in order to satisfy Eq. 1-61. The reduction of Hamilton's principle to a set of differential equations is best done using :f the calculus of variations. Therefore, a short digression into the calculus of variations is a worthwhile next step. ,}

1.3.1 Calculus of Variations

The calculus of variations is concerned chiefly with the determination of maxima and minima (more exactly extrema) of expressions involving unknown functions. It differs from differential calculus in that the variables are known in ordinary calculus and a minimum or maximum of a . function of these known variables is desired. In the calculus of variations,'1 the variables are unknown and it is desired that the relationships among { the variables be found which will form an extremum (e.g., a maximum. or minimum) of some integral containing these variables. I

The essential techniques of the calculus of variations can be determined by procedures analogous to those of finding maxima and minima of j differential calculus. As an example, consider the function I defined as~

'";1

the integral between 11 and t2 of another function L, where L is a function1i of the unknown variables q(t) and !J(t) and of the independent variable t:$

' ,1= 2 L(q(t); ,jet); 1) dt (1-62)J

'I

A typical problem is to find the function q(t) and also (j(l) which will

h,r " Illor.: delaile,1 treatmcnt of Ihe I:alculus of v;lriations see, e.g., F. B. Hildebrand, Methods oj Applied Mathematics, Prentice-Hall. New York, 1954, Chltp. 2.

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 33

make the integral I an extremum" and which will satisfy the end conditions

that q(ll) - q\ and q(12) - q2 (1-63)

where q\ and q2 are fixed end points. As a first step in finding the extremum of I assume that qo(t) is the

actual function which makes I an extremum, and choose any continuously J differentiable function 7)(t) which vanishes at the end points q(t\) = ql

~l

and q(lz) = qz. Then for any constant a, the function

q(t) = qo(l) + a 7)(1) (1-64)

II will satisfy the end conditions of Eq. 1-63. The integral/:

!

~ \, " I(a) == ". L(qo(t) + a 1)(t); 40(1) + IX >j(t); t) dl ( 1-65) rI

J

11

" ~ .', is obtained from Eq. 1-62 by setting

~ ~ q(t) = qo(t) + IX 7)(t) (1-64) '1\'

and 4(1) = 40(1) + IX 7j(/) (1-66)

The function l(a), Eq. 1-65, is only a function of lX, once qo(t) and 1](t) are assigned, and furthermore, the function I(a) is an extremum when a = a because qo(l) was chosen to make it so. However, this is only possible if

.1

dljda = 0 when ex = 0 (1-67) . J" \ (If>: Equation 1-67 is, therefore, the condition for an extremum and is a

.rJ' defining relation which relates differential calculus to the calculus of

'I;f.!,, variations. , . , The condition for an extremum, Eq. 1-67, can now be applied to Eq.

;': :}" 1-6'5 to find the differential equation which will result. Since a, in'".. ~, Eq. 1-65, is a constant in the integration. the differentiation wi.th respect ~.

to a can be taken under the integral sign to yield it

dl(a) II2 [OL () 8L,( )] d 0--= -1]1+---;;-7)11= (I -68) da 'I oq oq

Before investigating Eq. I-68 further, it is desirable to introduce the variational notatilln of the calculus of variations. The only variation to be studied here is the time-independent variation.

To tix idcas

II

I

:

ELECTR.OMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 3S34

Now return to the condition for an extremum, Eq. I-M!, and note that1.3.2 Time-Independent Variations (0 Variations) from the definition of oL, Eq. 1-74, ,

Consider the case in which times 'I and 12 of the end points are held I fixed and in which the integrand (L in Eq. 1-65) is an explicit function of j a dl(a) = J" 8L dt ( 1-75) the variables q, q, and'. Definc a variation of the function L in terms of cia '1 variations of q and q and not of the time. This is a time-independent Since u. is a constant in the integration with respect to I, the expression of (8) variation. Eq. 1-75 mu~t equal zero if Eq. 1-68 equals zero. Thus, the condition

The function qo(t) has been defined as the true function which produces for an extremum is an extremum, and any other function with the same end points was \ I' defined by Eq. 1-64 as 2 oLdt = 0 (1-76) (1-64) '1q(t) = qo(t) + a 7)(/)

Since the vanatlon is time-independent, it can be taken outside theNow define the variation oq as the variation of the function q(/) from the .~: integral sign to yield

true function qo(t) and obtain: ~,~

""!~,

8q = q(t) - qo(t) = a 7)(t) (1-69) Sf = S J'2L dt = J/2 SL dt = 0 (1-77)

'1 '1

Similarly, the variation oq of the velocity q from the true velocity 40 is Expanding SL by Eq. 1-74, the condition for an extremum in variational

(1-70)=oq = q(t) - 40(1) a 1j(t) notation can be expressed as

The next step is to define the variation of a function, e.g., the variation ' 2 (8L f:JL.)81 = a Sq + 8"'" 8q dt = 0 (1-78)of the function L(q(/); .:j(/); I). To do this find the difference which results J'I q q from a small variation in q(/) from the true function qo(/). This difference

Several forms of the condition for an extremum expressed in variationalis notation are given by Eqs. 1-76, 1-77, and 1-78. An extremum of theoL = L(q(t, a); q(t, a); t) - L(qo(t); qo(t); t) (1-71) function f in the calculus of variations is found by setting the variation of

where q(l, a) and 4(/,

36 ELECTROMECHANICAL ENERGY CONVERSION

(1-80)

for i = I, 2, 3, ... ,N. Since the variation 0 has been defined to be independent of the time f, it can be taken under the integral sign as in

Eq. 1-77 to yield

oj = J'2 SL(qt(f), ... ,qN(/); eMf), ., ., tiNCt); I) dl 'I

.,1'\

IlJ J/2!1: (8L Il 8L 8') 1 = .L. a qi + Y lJj C f = II ,-I q/ q,

Expanding the oL in Eq. 1-80 gives

0 (1-81) if

To simplify this expression, integrate the second term by parts. ovariation is independent of time; consequently,

The

oq = ~ oq (1-82)

and the second term of Eq. 1-81 can be integrated by parts to obtain

.~, 5 ' 2 ~ 8L 0'. d.:- ',' q, t 111

38 ELECTROMECHANICAL ENERGY CONYERSION t EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 39

IA State Functions II For example, it means that a knowlcdge of onc type of physical system may prove helpful in gaining an insight into a physical system of

anII entirely ditTerent nature. It also means that an engineer, whose work

In the previous sections the Euler-Lagrange equation of motion was

takes him outside his field of specialization, nced not fecI in completely

developed from Hamilton's principle; and, in so doing, the Lagrangian

strange territory.state function was introduced. The Lagrangian and the other sta

te

functions are of central importance in the characterization of physical Unfortunately, the class of systems exactly describable by state fun

ctions

docs not include all electromechanical systems. Dissipation must be

systems (electrical, mechanical, electromechanical, chemical, thermo-

excluded from systems if they are to be described by state functions. A

dynamic, etc.). The state functions include the total energy of the system

form of dissipation which proves particularly troublesome is hysteresis.

and other closely associated functions such as the Lagrangian. These

At first, this appears to be a severe limitation; however, the main use of

functions are called state functions because, at a given instant of time,

they depend solely on the state of the system at that instant of time, and

i.' i. (,not on past history. Their importance has been recognized for a lo

ng -'-- -, -+--+ Lossy + Loss lesstime in thermodynamics and in the statistical and quantum mechanical v' +

I electrical Vi electromechanical x,treatment of atomic systems, although their first noteworthy use was

in system - system

advanced classical dynamics. These state functions, and the variables

- :,~". ," For example, to describe a thermodynamic system, the I:

,'

"\.

Fig, 1-8. Lossy electromechanical system divided into simpler component parts.describing them, are in many cases used by engineers

without explicitly ..;""~,..'." realizing it. heating engineer will use such variables as temperature and entropy; th

e ,,.. state functions will be to obtain a general formulation of the equations of , -,;1

control system engineer will use force and displacement to describe his

I

connected mechanical system; the aeronautical engineer will talk about -

~.~

...~ motion of a system. The quantities which are most difficult to obtain

are the coupling terms between different types of systems, e.g., the

the roll torque and roll angle in discussing the stability of an aircraft; the

electromechanical coupling terms in eJectromcchanics. Fortunately,

electrical engineer will use voltage and charge for describing electric l..,~

these terms arc determined by the conservative part of the system and

circuit behavior; and the chemical engineer will employ such terms as ,'.r.

Each of these engineers works in [ ,. are derivable from state functions. Thus, it becomes practical to separate

chemical potential and mole number. the problem into two parts, consisting of (I) an energy conversion p

art his field of specialization, talking about these widely different physic

al For an

systems in terms which seem equally unrelated. In actuality, these that is dissipationless and (2) other parts with dissipation.

>'loR;' electromechanical system, this takes the form of a lossy electrical system

physical systems have much in common. (including hysteresis losses), a lossless electromechanical system, and

a For example, in each of the above cases denote the first variable by

It is possible in this way

11 and the second variable by qj; then, irrespective of the nature of the .,11r' lossy mechanical system as shown in Fig. 1-8. to study the lossless electromechanical system and bring in dissipationsystem, it is always possible to write l' ,i. when considering the over-all system performance. dW =/; dql (1-86)

,f;L}I;',; , 1.4.. 1 The Characterization of Physical Systems (Without Hysteresis where dW represents a differential change in energy produced by a t'~l' .},'. ~,',:,::,-;.. and Dissipation)

The variables f; and q, aredifferential change dqj in the variable q;. '.'

generalized vatiables and their product describes an energy relation in The constitution of a physical system from a dynamic point of view can

each of the above systems. The energy functio,ns may be, and usually be regarded as consisting of a number of particles, subject to inte

r- 'I

are, state functions and contain much valuable information about the connection and constraints of one kind or another. The configurati

on

system described by them. Actually, Eq. 1-86 is a manifestation of the of a given system at any time can be specified in terms of quantities call

ed

fact that in spite of their vastly different natures all physical systems have the coordinates of the system. The choice of the set of coordinates f

or

a fundamental similarity and lend themselves to a common mathematical a system is usually somewhat arbitrary, but in general each individu

al

This fact has far-reaching consequences for the engineer. energy storage element of the system can have a set of coordinates. For

description.

-41

40 ELECTROMECHANICAL ENERGY CONVERSION

example, every discrete element of mass can have its position specified in terms of three space coordinates, each inductance element can have its nux linkage specified, or each capacitor can have its total charge specified., Examrk's of possible coordinates for several systems are shown in Fig. 1-9. When dealing with static systems (systems in static equilibrium) the values of the coordinates completely specify the system. For a dynamic system, however, the coordinates do not completely specify the system and. an additional set of dynamic variables equal in number to the coordinates must be used. These dynamic variables can be the first derivatives of the coordinates, the velocities, or they can be a second set of variables, e.g., the momenta. The velocities and the momenta are associated variables and either set can be chosen as the dynamic variables.

So far, only the number of variables that can be ascribed to a particular system have been discussed; however, in any given system all of these variables may not be independent and hence they cannot all be specified independently. The question of how many variables ;ire independent is determined by the constraints of the system. The problem of handling the constraints is one of the most diflicult single questions of dynamics. Constraints are of two essential types-holonomic and nonholonomi~

constraints. The holonomic constraints are represented by sets of relations among the coordinates or, if expressed as differentials, they can be integrated to yield these relations. For example, if 11 coordinates can be ascribed to a system and then 111 equations of the form

.~(q\> ... ,q,,: t) = 0 j = I, ... , m (1-87)

can be written, it is possible to reduce the number of coordinates from n to (n - 111) by using these rn constraint equations to eliminate m variables. Holonomic constraints are always expressible in the form of Eq. 1-87; furthermore. for a system which has only holonomic constraints it is always possible to select a set of independent coordinates which does not contain the constraint equations. Thus, if n is the number of coordinates determined from all energy storage elements and m is the number of holonomic constraints, then there are (n - m) independent cr~rdinates

.. and (n - Ill) velocities, or a total of 2(n - m) variables which can be used to describe uniquely the dynamic motion of the system. The minimum value of (n - m) that can be found is also the number of degrees of freedom of a holonomic system. When a system is described, using a selected set of coordinates which eliminates the various system con-straints, it is accepted practice to caIl these coordinates the generalized coordinah:s of the system. For a system of N = (n - m) degrees of freedom there will always be 2N generalized variables needed to describe the dynamic path of the system (i.e., N coordinates and N velocities).

EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS

m ~

Adiabatic vwalls L e

s T

GasJiJ q, '" II = charge on C

q I Xl =displacement 01 mass q2 -J;' dt. !Otegral .Ofq, .. S .. entropy q l %2 = displacement of sprlOg Curren! In I.

qz =V =volume OR ({II (b) '1 X=I!ux I1nkmg L

q2 Z.,JF;dt =. Integral of voltage on C

(e) ,

c'OD~

q,. Q, Chafge on (.', :1'1 q2:= Q2:=: Charge on C

2 qJ fit dt :Il integral of current in L 1 q. ji2dt =integral of current in L 2

OR q 1=X -= dIsplacement

q\ '" A, tOl.1 flux linking L \ q2" Q:o Charge on C

q2 - .\.2 total flux linking L 2

q3f;, dt Integral of current in Lq, ~ fe, dl -_,ntegral of voltage .cross C\ (t)

q4 - fi2 dt integral of voltage across C2 (d)

,.~

I'r.:

~! Vr .~"l ?

~~

q 1 - :c 1 displacement of M C L q2 '" "'2 ~ displacement of spri",

Q3'" A" flux linking L q .. j'.dl .. integral volt.ge on C

Iron

(f)

Fig. 1-9. Examples of physical systems and variables ql associated with each energy storage, sources and dissipation omitted for simplicity.

~I"'

"

L

42 ELECTROMECHANICAL ENERGY CONVERSION EQUATIONS OF MOTION OF ELECTROMECHANICAL SYSTEMS 43

The questions of independence of coordinates which arise when there required and can be expressed as ql(/), qz(/) . ... , qN(I) where the q;(t)

ex.ist nonholonomic constraints are much morc dil1icult to resolve. A are the genl'l"u/i::cd coordinafes. Systems so described are calkd dynami

c

For dynamic systems a second set of N quantities can benonholonomic constraint is one in which equations of constraint do not systems.

ex.ist among the coordinates, i.e., constraints not satisfying Eq. 1-87. introduced such as the PI(/). P2(t), ... , PN(I). called the genera/b,

d

The slale of a dynamic systcm at a Ricen inSlanl of lillie isFor example, a nonholonomic constraint is obtained when the gcncral momenta.

determined by the particular values or the N generalized coordinates and

form of thc constraint equation is Thus the statc or athe N generalized momenta at that instant or time. i((/I> ... , 4n; t) = 0 (l-88) dynamic system may be represented as a point in a 2N-dimensional

where the resulting differcntia1 cquation is not intcgrab1e. An example

of such a nonholonomic constraint is the commutator in electric machines

por the constraint imposcd on a rolling ball by a perfcctly rough surface

.

Another type of nonholonomic constraint is the inequality type of

constraint such as that imposed by the wall of a container upon a gas

", r

I,

particle contained therein. In this case the nonholonomic constraint is At'..1,of the form

(ql)2 - b2 < 0 (1-89) " ",

where ql is the coordinate of the gas particle and b is the coordinate of ".;(""the wall of the container.

"\ .When dealing with nonho10nomic constraints it is not possible to find " ... % 116I Xoa set of generalized independent coordinates equal in number to the ~.',

number of degrees of freedom. Instead, it is necessary to choose a

number of coordinates equal to the number of degrecs of freedom plus

the number of nonho10nomic constraints. For problems of this type the

Euler-Lagrange equation derived in Sec. 1.3.3 from Hamilton's principle

cannot be used since the coordinates are not independent. In general,

\,'>"i J any problem with nonholonomic constraints is very difficult to handle

unless some trick can be devised to reduce it to an cquivalent holonomic ~i.l 1

problem. A mcthod for doing this will be discussed latcr in Sec. 1.6. ~

":.Fig. 1-10. Path in phase space for system of Fig. 1-9a

when p = 0 and x = ."0 at I = O. 1.4.2 Generalized Coordinates for Holonomic Systems

There are two types or states of physical systems: static and dynamic. space, the 2N dimensions being the N coordinates ql and the N momenta

For the static state, only the description of the system in static equilibrium

PI' Thic ~pace is called phase space. Once the state of a conservative

with its environment is given. The state of a static system is completely f

system is established at one time its path in phase space is completely

specified by the values of its N generalizcd coordinates. For a system in determined. This means that oncc a given set of ql(ll) and 1'1(11) is estab- .,static equilibrium there can be no dissipation. In discussing stati

c lished, then ql(l) and